Is it possible to animate a single mesh as if it were multiple objects?
I don't think it's possible to add more than one armature to a single object (haven't tested it, though).
What you can do is create all three bones inside a single armature and then parent the mesh to the armature with automatic weights. I think it'll weight properly, but if it doesn't just manually edit the weights so all the vertices in each box are only affected by its respective bone.
I don't know if that helps for your specific needs.